The Key Idea
An accumulation function is a function defined using a definite integral. A common form is
At first glance, finding where this function has a maximum or minimum may look difficult because the function is defined by an integral rather than by an ordinary algebraic formula.
The Fundamental Theorem of Calculus makes the problem much easier. If the upper limit is , then differentiating the accumulation function gives the integrand evaluated at :
That means an extrema problem involving an accumulation function can often be solved exactly like any other extrema problem: find where the derivative is zero, determine the sign of the derivative, and classify the critical points.
If , then . You usually do not need to evaluate the integral first. Find the critical points by setting the integrand equal to zero.
Why the Integrand Controls the Extrema
Suppose
By the Fundamental Theorem of Calculus,
Therefore, the sign of is determined entirely by the sign of .
- If changes from positive to negative, then has a relative maximum.
- If changes from negative to positive, then has a relative minimum.
- If does not change sign, there is no local extremum at that critical point.
A Step-by-Step Method
- Identify the accumulation function. Look for a function defined by an integral whose upper limit contains the variable.
- Differentiate using the Fundamental Theorem of Calculus. The derivative is the integrand evaluated at the upper limit, multiplied by the derivative of the upper limit if the upper limit is something other than simply .
- Find critical points. Set the derivative equal to zero or identify where it is undefined.
- Make a sign chart. Determine whether the derivative changes from positive to negative or negative to positive.
- Classify the extrema. Positive-to-negative means a maximum; negative-to-positive means a minimum.
- For absolute extrema, compare endpoints. A relative extremum is not automatically an absolute extremum on a closed interval.
Worked Examples
Let
Find the value of where has a relative maximum.
Let
Find the relative extrema of .
Let
Find the critical points of .
Let
Find the absolute maximum of on .
Relative vs. Absolute Extrema
This distinction matters frequently on AP Calculus questions.
A relative maximum only compares a point to nearby points. An absolute maximum compares it with every point in the specified interval.
For a continuous function on a closed interval , the Extreme Value Theorem guarantees that an absolute maximum and absolute minimum exist. To find them, check:
- All critical points in the interval.
- The left endpoint.
- The right endpoint.
To find where an accumulation function has an extremum, you generally only need its derivative. The Fundamental Theorem gives that derivative immediately. You only need to evaluate the original integral when the question asks for the actual value of the function at the extremum.
Common Mistakes
Extrema occur where the derivative is zero, not necessarily where the accumulation function itself is zero. If , look at when searching for critical points.
If the upper limit is rather than simply , you must multiply by . This is the chain rule applied to an accumulation function.
A critical point is only a candidate. You must check the sign of the derivative. If the derivative stays positive or stays negative on both sides, there is no relative maximum or minimum.
A relative maximum does not automatically give the absolute maximum on an interval. For an absolute-extrema question, always check the endpoints as well as the critical points.
Practice Problems
Let . At what value of does have a relative maximum?
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Let . Find the relative minimum of .
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Let . Find all critical points of .
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Let . Find the absolute maximum of on .
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